Peter Hertel. University of Osnabrück, Germany. Lecture presented at APS, Nankai University, China.

Transcription

1 s s University of Osnabrück, Germany Lecture presented at APS, Nankai University, China Spring 2012

2 s Structure your program into Scripts are unnecessary and should be avoided more on are objects they may be arguments of! example: integral more on function, or functionals example: the bisection algorithm

3 s Input and output values In mathematical language, a function y = f(x) maps a real valued variable x into a real valued variable y In Matlab the variable x has to be represented by a finite number of representative values such as x=linspace(0,2*pi,1024); which creates a row vector with 1024 equally spaced values in [0, 2π]. Endpoints are included. y=sin(x); returns the sine values of the representative values. Matlab may return more than one output value they must be grouped by [...] such as in [c,s]=cossin(x); Matlab may depend on no, one, or more arguments. Matlab may return no, one, or more results.

4 s function [c,s]=cossin(x) c=cos(x); s=sin(x); end % function cossin >> x=linspace(0,2*pi,1024); >> [a,b]=cossin(x); >> plot(x,a,x,b); >> axis tight

5 s Our first plot: cos(x) and sin(x) vs. x.

6 s A function fun is a piece of software. Either defined in an.m file, or built-in. It resides somewhere in RAM The RAM address of the function is means located at as is well known from addresses It is not the object itself, but a handle how to access the object for example, s=@sin says where the sine function is located in memory. y=sin(x) and y=s(x) return the same result, namely the sine of x. s can be variables themselves.

7 s Anonymous you may declare by their handle only 1./(1+(a*x).^2) the current value of a is used note the dots the handle has a name, here fh the function itself has no name (anonymous)

8 s als A functional is a function which depends on a function e.g. the integral quad(f, a, b) = b a dx f(x) quadrature (of the circle) in Matlab: q=quad(f,a,b) f must be a function, or anonymous >> f=@(x) sqrt(1-x.^2); >> pa=4*quad(f,0,1) pa =

9 s Squaring the circle About 1500 BC the Chinese used the modern decimals system About 450 BC, the Chinese mathematician and astronomer Zu Chongzhi approximated π by 355/113 This is correct for first seven digits About the same time, in ancient Greece, mathematicians (philosophers) tried to find an exact expression : reshape a figure so that it has the same area until it becomes a square, or a quadrate The Greek mathematician never reached their goal because this is not possible in a finite number of steps Only modern mathematics introduced infinite series and their limit values The integral is such a limit.

10 s Thabit ibn Qurra lived (arond 800 AD) and taught in Baghdad, today Irak. He invented a very elegant proof of Phythagoras s theorem.

11 s b c a By moving equally sized triangles, Thabit proved a 2 + b 2 = c 2.

12 s Accuracy A computer has only a finite number of states and a finite time to perform a calculation e.g. numbers like 2 or π are never exact Since a computer uses the binary number system, even 1/3 will be approximated therefore the quad function can calculate the integral with a finite accuracy only default is five digits this can be improved pa=4*quad(f,0,1,1e-12) is accurate for 11 digits note that quad can be called with three and with four arguments

13 s ode45 solve the ẏ = f(t, y) first, specify the by a function f = f(t, y) specify the time span tspn specify the initial condition y 0 [tout,yout]=ode45(@f,tspn,y0) does the job example: standard logistic curve ẏ = y(1 y) currently, the market share is 10%

14 s % calculate the standard logistic curve % for a given initial condition % setup the ydot=@(t,y) y.*(1-y); % time span ts=linspace(0,4,512); % initial condition y0=0.10; % solve it by ode45 [t,y]=ode45(ydot,ts,y0); % plot it plot(t,y); axis([0,4,0,1]); % export it print -deps slc.eps

15 s The standard logistic curve as solved numerically by ode45.

16 s algorithm Let f be a continuous function and [x 1, x 2 ] an interval. If f(x 1 ) and f(x 2 ) have different sign, then there is an x [x 1, x 2 ] such that f(x) = 0. In Matlab, f is a function handle. >> pa=2*bisection(@cos,0,2,1e-12); >> pa-pi e-13

17 s function x=bisection(f,x1,x2,tol) maxit=50; f1=f(x1); f2=f(x2); if f1*f2>0 error( bad initial interval ); end; it=1; while (abs(x2-x1)>abs(tol)) && (it<maxit) it=it+1; xc=(x1+x2)/2; fc=f(xc); if f1*fc<0 x2=xc; f2=fc; else x1=xc; f1=fc; end; end x=(x1+x2)/2; end % bisection